If n is a one-digit positive integer, what is n?
(1) The units digit of `4^n` is 4.
(2) The units digits of `n^4` is n.

Answer

To solve the problem, we need to determine the value of \( n \), a one-digit positive integer, based on the two statements provided.

### Step 1: Analyze Statement (1)
The first statement tells us that the units digit of \( 4^n \) is 4.

To find the units digits of \( 4^n \) for one-digit positive integers, we can compute the first few powers of 4:
- \( 4^1 = 4 \) (units digit is 4)
- \( 4^2 = 16 \) (units digit is 6)
- \( 4^3 = 64 \) (units digit is 4)
- \( 4^4 = 256 \) (units digit is 6)
- \( 4^5 = 1024 \) (units digit is 4)

From this, we can see a pattern:
- The units digit of \( 4^n \) is 4 when \( n \) is odd (1, 3, 5, 7, 9).

Thus, from Statement (1), we conclude that \( n \) can be:
- \( 1, 3, 5, 7, 9 \).

### Step 2: Analyze Statement (2)
The second statement tells us that the units digit of \( n^4 \) is \( n \).

Now we will check the values of \( n \) from the first statement against this condition:
- For \( n = 1 \): \( 1^4 = 1 \) (units digit is 1)
- For \( n = 3 \): \( 3^4 = 81 \) (units digit is 1)
- For \( n = 5 \): \( 5^4 = 625 \) (units digit is 5)
- For \( n = 7 \): \( 7^4 = 2401 \) (units digit is 1)
- For \( n = 9 \): \( 9^4 = 6561 \) (units digit is 1)

Now we summarize the results:
- \( n = 1 \): units digit is 1 (does not satisfy)
- \( n = 3 \): units digit is 1 (does not satisfy)
- \( n = 5 \): units digit is 5 (satisfies)
- \( n = 7 \): units digit is 1 (does not satisfy)
- \( n = 9 \): units digit is 1 (does not satisfy)

From Statement (2), the only value that satisfies both conditions is \( n = 5 \).

### Conclusion
Thus, the solution to the problem is:
\[ n = 5 \]